Thread: Proof for average velocity

1. Proof for average velocity

Hi all,

I want to prove the following algebraically (that is, NOT with calculus): If an object with initial velocity $\displaystyle v_i$ is accelerated at some constant acceleration over a time $\displaystyle t$, and finishes with a final velocity of $\displaystyle v_f$, then the average velocity of the object is $\displaystyle {v_i + v_f} \over 2$. This average velocity is the equivalent velocity that this object would have to be travelling (without acceleration) to travel the same distance as the object experiencing acceleration as above.

I need to somehow prove this in order to derive the kinematic equations. But how? It is easy to derive the kinematic equations with a knowledge of calculus, but if you are teaching this to someone without a knowledge of calculus, how can this be proven rigorously?

2. Re: Proof for average velocity

Originally Posted by borophyll
Hi all,

I want to prove the following algebraically (that is, NOT with calculus): If an object with initial velocity $\displaystyle v_i$ is accelerated at some constant acceleration over a time $\displaystyle t$, and finishes with a final velocity of $\displaystyle v_f$, then the average velocity of the object is $\displaystyle {v_i + v_f} \over 2$. This average velocity is the equivalent velocity that this object would have to be travelling (without acceleration) to travel the same distance as the object experiencing acceleration as above.

I need to somehow prove this in order to derive the kinematic equations. But how? It is easy to derive the kinematic equations with a knowledge of calculus, but if you are teaching this to someone without a knowledge of calculus, how can this be proven rigorously?
We have to choose a couple of equations that don't depend on your answer for the average speed. (There's no real difference between velocity and speed in this case. I'm just picky that way.)

So first, always set up a coordinate system. I will choose an origin to be at the starting point of the measurement ($\displaystyle x_i = 0$) and final point to be at $\displaystyle x_f$. I am choosing the positive x direction to be in the same direction as the acceleration, and we might as well choose that to be horizontal and to the right.

Okay, now we need three equations and put them together. I'm going to choose:
1. $\displaystyle v_f^2 - v_i^2 = 2a(x_f - x_i) \implies v_f^2 - v_i^2 = 2ax_f$ (remembering that $\displaystyle x_i = 0$)
and

2. $\displaystyle v_f - v_i = at$

So we have for the average speed:
$\displaystyle v_{ave} = \frac{x_f - x_i}{t} \implies v_{ave} = \frac{x_f}{t}$

Solving equation 1 for $\displaystyle x_f$ and putting it into equation 2:
$\displaystyle v_{ave} = \frac{v_f^2 - v_i^2}{2at}$

I'll leave the rest to you. You need to factor the numerator, plug in $\displaystyle v_f - v_i = at$ and do some simplifying. If you need help to finish it, just ask.

-Dan

3. Re: Proof for average velocity

Hi topsquark,

The problem I have with this is that Equation 1 has not yet been derived yet. I know it is one of the common kinematic equations, but we cannot use this formula until it has been derived, if you know what I mean? And to derive it requires proving the formula for average velocity, sort of a chicken and egg problem!

At this point, all I have to work with is the formula $\displaystyle v_f = v_i + at$. I don't think it is possible to prove the average velocity formula without a knowledge of calculus, in particular integration and what the area under a curve means.

4. Re: Proof for average velocity

As I have said above, knowing that the area under the graph means total displacement implies a knowledge of calculus. I have come to the conclusion that the average velocity formula cannot be proved without a knowledge of calculus.

5. Re: Proof for average velocity

Originally Posted by cybertutor
i have replied to this thread twice now but it seems to be disappearing for some reason.
The way you need to do it is using graphs....area under velocity time graph is the distance and you can compare a trapezium shape with a rectangular which will give you an equation of average vs changing velocity.
Hope this helps.
I agree. I can't find a way to do it either.

-Dan

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deriving kinematic equations without calculus

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